In thermal processes and plasma processes in semiconductor manufacturing equipment, flat panel display (FPD) manufacturing equipment, or solar cell manufacturing equipment, there is the need to control important state quantities, such as the surface temperature of a wafer or a glass panel (the temperature of the actual object), or the like, online during the execution of the process. However, it is difficult to perform a process with a temperature sensor attached to the surface of the wafer or the glass panel.
Given this, the critical state quantities are controlled online through first investigating, off-line, the relationship between a temperature in a location that can be measured during the execution of the process and the surface temperature of a wafer or a glass panel (the temperature of the actual object) that cannot be measured during the execution of the process, and then to estimate the surface temperature of the wafer or glass (the temperature of the actual object) based on the relationship, understood in advance, to the temperature that can be measured during the execution of the process. In such a case, methods for calculating polynomials that linearly approximate the numerical relationships between the measurable temperature and the surface temperature of the wafer or the glass panel (and estimations of state quantities using polynomials) are widely performed through the application of multivariate analysis methods to measurement data (analysis data) for the temperatures that can be measured and the surface temperatures of the wafers or the glass panels (the temperatures of the actual objects), obtained off-line. (See, for example, Japanese Unexamined Patent Application Publication H5-141999.) When multivariate analysis methods are used, the temperature that can be measured during the execution of the process is positioned as an input parameter in the polynomial. On the other hand, the surface temperature of the wafer or the glass panel (the temperature of the actual object), which is the subject of the estimation, is positioned as the output parameter of the polynomial.
In many cases, that which is subject to state quantity estimation does not have a simple linear relationship between the input parameter and the output parameter. Consequently, if one wishes to improve the accuracy of the state quantity estimation then it is necessary to increase the order of the estimating polynomial that is calculated through the multivariate analysis. At this time, unless the input parameter is varied experimentally, there will be a scarcity of data in the parameter space on the input parameter side of the analysis data. When increasing the order of the estimating polynomial there is a tendency for the accuracy to increase in the region wherein the data are dense, but in region wherein data are scarce, there is a high probability that the resulting polynomial will calculate unrealistic estimated values. In particular, when estimating functions are built into equipment such as semiconductor manufacturing equipment and the equipment manufacturers ship the equipment to the equipment users, and the equipment users gather analysis data, the input parameter space envisioned by the equipment manufacturer side will not necessarily match the input parameter space as understood by the equipment user side. Consequently, regardless of the fact that that this problem with density/scarcity of the collection of analysis data tends to occur when this type of equipment is distributed, this problem tends to be overlooked.
In order to simplify the explanation, let us assume that there is a single input parameter. Let us assume that six combinations of values A through F have been obtained as analysis data with the (X, Y) combinations, of the input parameter X and the output parameter Y as follows: A (1.6, 20.024), B (2.0, 21.000), C (2.4, 23.304), D (2.8, 27.272), E (3.2, 33.288), and F (3.5, 39.375). At this time the distribution of the analysis data in the six combinations of A through F is as illustrated in FIG. 12.
While this is a distinctive feature of the analysis data at this time, let us assume that, in consideration of the physical relationship between the input/output parameters (X, Y) that intuitively one can expect there to be a monotonically increasing relationship. That is, one can assume that, by prior knowledge, one can envision the relationships between the input and output parameters (X, Y) to be as in FIG. 13. Even when there is such a relationship, situations wherein data cannot be obtained in the vicinity of X=0, that is, wherein there is a region of data scarcity due to circumstances on the equipment user side, such as an awareness of data collection, frequently occurs in workplaces such as in semiconductor manufacturing. When the combinations of the A through F data are used in, for example, multivariate analysis for a third-order polynomial in order to achieve high accuracy in reproducing the relationship of the input/output parameters (X, Y), then an equation such as the following will be produced:Y=X3−2.0X2+21.0  (1)
The third-order curve 220 illustrated in FIG. 14 is obtained from the third-order polynomial of Equation (1). On the other hand, 221 is a curve that illustrates the relationship between the input and output parameters (X, Y) obtained from common-sense assumptions such as described above. As illustrated in FIG. 14, the third-order polynomial of Equation (1) matches the data A through F with high accuracy. On the other hand, according to this third-order polynomial, at the point in the vicinity of X=0, S (0.0, 21.000) will result. That is, there are data-scarce regions everywhere other than the data-dense region of 1.6≦x≦3.5 in the parameter space of the input parameter X, and in these data-scarce regions the third-order polynomial of Equation (1) is a polynomial that produces unrealistic estimated values.
If this situation wherein unrealistic estimated values are calculated by the estimating polynomial in this way is overlooked, for example, if temperatures are estimated online in a semiconductor manufacturing process, then there will be a region wherein one can expect highly accurate estimates (the data-dense region), and regions wherein unrealistic estimates will be made (the data-scarce regions). Given this, there has been the possibility that there can be a large deleterious impact on the manufacturing process in the regions wherein unrealistic temperature estimates are made.
The present invention was created in order to solve the problem set forth above, and the object thereof is to provide an estimating polynomial generating device, an estimating device, an estimating polynomial generating method, and an estimating method able to calculate an estimating polynomial that enables a reduction in the probability of the calculation of an unrealistic estimated value in a region wherein the analysis data are scarce, when performing estimates of, for example, state quantities using an estimating polynomial, wherein the estimating polynomial was calculated using the analysis data.